After 53 Ka3 the game is a win for Black. Black should play 53...b4+ 54 Ka2 Kc2 55 Ka1 Kb3 (not ...b3?? because of stalemate) 56 Kb1 Ka3 57 Ka1 b3 58 Kb1 b2 59 Kc2 Ka2 and the pawn promotes.
54...b4 is a blunder, and allows White to draw with the move you played, Kc2. Now, there is no way for Black to win, as White will never allow Black to gain the opposition to take control of the key squares a3, b3, and c3. Black had to play instead 54...Kb4, which takes control of those key squares. See Diagram.
In the game geert-seuntjes I met an interesting situation which resulted in a draw (although I intended to abandon first).
I had only my king and my opponent his king and a pawn.
When his pawn was on b3 and his king behind it, I realized I could make a draw of it. When his king was on the 4th line I needed to stay on the b1 and b2 field, when he moved to the third line, I had to face him.
What is more interesting is that I think that if the king of my opponent had preceded his pawn, he would have won. He made a mistake on move 54.
Is my analysis correct?
geert